What are the rules for marble solitaire? | Reference.com What are the rules for marble solitaire? A: Quick Answer In marble solitaire, the player jumps marbles over each other into existing spaces with the goal to eliminate all the marbles from the playing board except one. Most marble solitaire game sets contain playing boards with 33 spaces and 32 marbles, which means 31 marbles must be eliminated to win the game. Full Answer Also called peg solitaire when played with wooden or plastic pegs, marble solitaire boards look different depending on the geographical origins of the game and the manufacturer. Marbles can be arranged in various shapes, including crosses, circles, diamonds, triangles and stars. The player sets up the board by filling all of the divots with marbles except for the center space. They start game play by jumping one of the marbles into that central divot. The player can only jump into an empty space, and they cannot jump multiple marbles at once. When a marble is jumped, that marble is removed from the board. Game play ends when the player is not able to make another move. This usually occurs because the remaining marbles are too far away from each other to be jumped.

The player scores their game by counting the remaining marbles on the board. Peg Solitaire Peg Solitaire What is Peg Solitaire? Peg solitaire is an old board game for one player. A game board with 33 holes in cross form is given. 32 pegs are in. The centre hole is empty. You have to remove the pegs one after the other by jumping horizontally or vertically over one. In the end one peg should be left in the centre hole. The game is called solitaire (Latin solus = alone) because of the only player. The game solitaire is called jumper, too. The name is not unique. A solitaire can also be a card game for one player or a piece of jewelery having a single diamond. Removing all pegs but one is not easy to solve. ... Generally you play this game without a strategy, so that you can't repeat the moves later. If you like to tell a solution, you have to establish a notation scheme. E.g. you can number the holes in lines. "15-17" means you jump peg 15 to 17 and remove peg 16.

1st solution: 15-17, 28-16, 21-23, 07-21, 16-28, 31-23, 24-22, 21-23, 26-24, 23-25, 32-24-26, 33-25, 26-24, 12-26, 27-25, 13-27, 24-26, 27-25, 10-12, 25-11, 12-10, 03-11, 10-12, 08-10, 01-09-11, 02-10, 17-05, 12-10, 05-17. You need 31 jumps, because you must remove 31 pegs.

Double jumps like (32-24-26) or (01-09-11) counts as one move. So the solution above has 29 moves. The board is symmetrical horizontally, vertically or diagonally, therefore there are seven more solutions. 2nd solution: The "world record" is a solution with 18 moves by E. Bergholt from 1912. 15-17, 28-16, 21-23, 24-22, 26-24, 33-25, 18-30, 31-33-25, 09-23, 01-09, 06-18-30-28-16-04, 07-21-23-25, 13-11, 10-12, 27-13-11, 03-01-09, 08-10-12-26-24-10, 05-17.

3rd solution: The following solution is clever. You use the L-move four times and get the house figure (the house stands on its head). Then you take down this figure with a six-jump move. ... ... 05-17, 08-10, 01-09, 03-01, 16-04, 01-09, 28-16, 21-23, 07-21, 24-22, 21-23,26-24, 33-25, 31-33, 18-30, 33-25, 06-18, 13-11, 27-13, 10-12, 13-11, 24-26- 12-10-08-22-24, 17-15, 29-17, 18-16, 15-17. Mobility of the Pegs top .. ...

If a peg is in the middle (red), jumps in all directions are possible. Two jumps can be made from the blue field and one jump from the green one. There are 76 jumps altogether. The moves on the board are more complicated because you don't know which moves will be used or which ones will be repeated in the same place. ... ... You can make another classification of the holes of the field. There are four classes (4 colours). A peg always stays in the same class. This classification is very important in the theory of solitaire and play of solitaire. (See more in book 2). A Pattern at the Beginning top Getting rid of as many pegs in patterns as possible is a simple way of playing solitaire. One peg must be left over somewhere. These simple puzzles are good for beginners to learn to play the big solitaire. Solutions: Steckhalma.exe und SteckhalmaLösung.ebt . Building of a Solitaire Game top There is hardly a household in my surrounding, which has no solitaire game. Nevertheless I will shortly explain, how to build a game. (Drawing 1) In the simplest case you draw the game pattern on paper and use any tokens.

(Drawing 2) This is the standard version of the game. You draw the cross pattern on a thick piece of wood and drill holes with the same depth. You succeed in making the same holes with a "depth feeler" on the drill. Then you must produce pegs with the same length (or you take dowels), which must lightly stick in the holes. (Drawing 3) You can drill through the board. Then you must make pegs using two dowels of different diameters. (Drawing 4) You can drill hemispheres (or half rounds) and lay marbles in it. Other Boards top The game with 33 holes described up to now is the standard game at least in Germany. It is called the English solitaire. French Solitaire Mathematische Unterhaltungen und Spiele, Band I. Leipzig, 2.

Aufl., 1910, pp 182-210 - E R Berlekamp, J H Conway, R Guy; Winning Ways, for your mathematical play, Vol 2, (Part 4: Diamond) Academic Press, London, 1982 chapter 23 Puring Pegs Properly, pp 697-734 (German: Gewinnen, Strategien für mathematische Spiele, Band 4: Solitairspiele, Vieweg Verlag, Braunschweig, 1985) - John D. Beasley, "The Ins & Outs of Peg Solitaire", Oxford Univ. Press, 1985 (paperback Edition 1992, contain an additional page: Recent Developments) ISBN 0-19-286145-X (paperback) An application of elementary group theory to central solitaire. College Mathematics Journal. (May 1998) v. 29(3) p. 208-212.

ZblD 1999b.01301 A Solitaire Game ans Its Relation to a Finite Field, Journal of Recreational Mathematics (1972) v. 5(2) p. 133-137 - Pieter van Delft, Jack Botermans; Denkspiele der Welt, (orig: Creative Puzzles of the World) - Jörn Eichler, Jochen Jäger, Thomas Ludwig; Spielverderber - Solitaire mit dem Computer l\"osen, c't 1999 Heft 7, 218-225 - Martin Gardner, The Unexpected Hanging and Other Mathematical Diversions Simon & Schuster (1968) Chapter 11: Peg Solitaire. p122-135 the 33-hole solitaire and the 6-square.

(Logik unter dem Galgen, Vieweg, 1971, Kapitel 11: Patience mit Figuren) - Kurt-Heinz Grube; MU (Der Mathematikunterricht) 26:2 (1980) 37-63 ZblD 1980x.01194 - Reiss' 'three-pile-game' , explains the 16 fundermental classes - solves several special figures and the standard problem. - Irvin Roy Hentzel; Journal of Recreational Mathematics (1973) v. 6(4) p. 280-283 - Irvin Roy Hentzel, Robert Roy Hentzel; Triangular Puzzle Peg, Journal of Recreational Mathematics (1986) v. 18 p. 253-256 - solving the order 5 triangle - E. Lucas; 2nd edition, 1891, pp 87-141 - Miller, W.A.; Triangular peg solitaire on a microcomputer. Journal of Recreational Mathematics (1991) v. 23(2) p.

109-115. ZblD 1992d.00202 33 Hole Peg Solitaire 33 Hole Peg Solitaire How many ways (number of solutions) are there to win the 33 hole (32 pegs) version of Peg Solitaire? (Click here for 37-hole Peg Solitaire ) Rules of the game: The 33-hole version of peg solitaire consists of 33 holes (see diagram below) and 32 pegs. To start the game, a player places the 32 pegs in the holes leaving the center hole empty. (The game may also be played by simply drawing the diagram on a piece of paper and then using any 32 markers as the pegs.) Then moves are made by taking any peg, jumping over another single peg and landing in an empty hole. Moves may be in any horizontal or vertical direction but must be in a straight line. Each jumped over peg is removed from the board. As an example of a legal move, the starting position has a hole at position 16, and all other holes are filled.

The 4 possible legal moves are: 4 over 9 to 16 (remove the peg at 9), 14 over 15 to 16 (remove the peg at 15), 18 over 17 to 16 (remove the peg at 17), or 28 over 23 to 16 (remove the peg at 23). If a player can make 31 moves and leave the last peg in the center hole, the player wins. 0 1 2 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 A few questions can be raised at this point: How many different games are possible? How many ways are there of winning? What is the shortest (worst) possible game (no more legal moves)? Can the game be won using some other hole as the starting and end point?

First, there are 577,116,156,815,309,849,672 different game sequences. That's: 577 quintillion 672 possible game sequences of Hi-Q. From this total, the number of solutions is 40,861,647,040,079,968. (Most of these are rotations and reflections of unique solutions).

If you want to express this as a number, it is: 40 quadrillion 968 Sample solution: Using the board notation given above, one way to solve the puzzle would be the following sequence: 4->16, 7->9, 0->8, 2->0, 9->7, 6->8, 10->2, 12->10, 15->3, 0->8, 13->15, 15->3, 17->5, 2->10, 19->17, 17->5, 27->15, 20->22, 22->8, 3->15, 15->17, 24->10, 5->17, 26->24, 23->25, 32->24, 17->29, 30->32, 32->24, 25->23, 28->16 Another solution is: 28->16, 9->23, 

18->16, 16->28, 31->23, 29->17, 26->24, 17->29, 32->24, 12->26, 23->25, 26->24, 5->17, 24->10, 11->9, 21->23, 30->22, 23->21, 20->22, 15->27, 13->15, 8->22, 27->15, 0->8, 2->0, 15->3, 6->8, 9->7, 0->8, 7->9, 4->16 Shortest (Worst) Possible Game It is possible to reach a dead end in just six moves using the following sequence: 4->16, 23->9, 14->16, 17->15, 19->17, 31->23 This leaves the following peg pattern: 

0 1 2 6 7 8 9 10 11 12 13 15 17 20 21 22 23 24 25 26 27 29 30 32 Number of Possible Positions The table below shows the number of positions that are reachable in the standard game.

(Start and finish at the center hole.) The count includes all possible positions – many of which are a rotation and/or a reflection of single unique position. (Earlier versions of this web page just counted unique positions.) Notes: Total positions are those reachable by ordinary game moves. Total Arbitrary Positions include all possible peg placements. Most of these positions are not reachable in a standard game. The number of Total Arbitrary Positions given “N” holes is equal to COMBIN(33,N).

Nbr Nbr Total Number of ways of of Total Winning Arbitrary to hit a dead end Holes Moves Positions Positions Positions on this move 1 0 1 1 33 0 2 1 4 4 528 0 3 2 12 12 5,456 0 4 3 60 60 40,920 0 5 4 296 292 237,336 0 6 5 1,338 1,292 1,107,568 0 7 6 5,648 5,012 4,272,048 32 8 7 21,842 16,628 13,884,156 0 9 8 77,559 49,236 38,567,100 0 10 9 249,690 127,964 92

,561,040 0 11 10 717,788 285,740 193,536,720 280 12 11 1,834,379 546,308 354,817,320 31,920 13 12 4,138,302 902,056 573,166,440 0 14 13 8,171,208 1,298,248 818,809,200 386,416 15 14 14,020,166 1,639,652 1,037,158,320 18,168,144 16 15 20,773,236 1,841,556 1

,166,803,110 52,363,776 17 16 26,482,824 1,841,556 1,166,803,110 569,426,456 18 17 28,994,876 1,639,652 1,037,158,320 36,408,754,040 19 18 27,286,330 1,298,248 818,809,200 380,028,309,224 20 19 22,106,348 902,056 573,166,440 8,520,659,218,816 21 20 15,425,

572 546,308 354,817,320 195,539,172,954,288 22 21 9,274,496 285,740 193,536,720 3,720,848,140,835,952 23 22 4,792,664 127,964 92,561,040 53,107,584,231,437,936 24 23 2,120,101 49,236 38,567,100 604,666,435,961,470,144 25 24 800,152 16,628 13,884,156 4,407,

068,360,590,383,432 26 25 255,544 5,012 4,272,048 21,625,765,333,357,588,280 27 26 68,236 1,292 1,107,568 82,475,526,111,015,358,616 28 27 14,727 292 237,336 135,521,776,905,015,042,336 29 28 2,529 60 40,920 210,993,770,838,107,635,688 30 29 334 12 5,456 1

05,088,038,911,515,040,128 31 30 32 4 528 16,260,787,716,385,283,832 32 31 5 1 33 81,723,294,080,159,936 Totals 187,636,299 13,428,122 577,116,156,815,309,849,672 Totals for move 31 consist of 40,861,647,040,079,968 ways to end in the center hole plus 4 times 10,215,411,760,019,992 for the other single-hole solutions.

= 40,861,647,040,079,968 + 40,861,647,040,079,968 = 81,723,294,080,159,936 Interpretation: The first line is the starting position for the game. There is only one pattern (hole in the center) and the solution can be found from this position. After move number 1, there are 2 empty holes. There are 4 possible board patterns and the solution can be reached from all 4 of these. (All 4 of these possible board patterns are rotations of a single unique pattern.) After 2 moves there are 3 empty holes. There are 12 possible board patterns and the solution can be reached from all 12 of these. After 4 moves there are 296 possible board positions, but only 292 of these can lead to a solution. After 30 moves (two pegs left on the board) there are 32 possible patterns, but only 4 of these lead to a solution.

Finally, if the game ends after 31 moves, the peg must either be in the center, or it must be in one of the 4 positions 1, 13, 19, 31. Alternate Games The game can also be played using other holes as a starting and finishing point. The table below shows the number of possible board combinations (includes the start position), the number of ways to win, and the number of possible games given the start hole. (Hole numbers use the 0 – 32 numbering system shown earlier.) Start/End Total Board Nbr.

of Ways Total Game Hole Positions to Win Combinations 0 207,684,279 2,343,652,440,537,181,612 1,547,384,243,264,761,654,653 9 206,218,425 8,940,989,276,947,390,168 3,214,909,287,232,785,028,120 16 187,636,299 40,861,647,040,079,968 577,116,156,815,309,849,672 The most difficult of the above variations is to start with a hole at position 1 and then leave the final peg at position 1.

A possible solution is: 9->1, 23->9, 18->16, 5->17, 24->10, 26->24, 12->26, 16->4, 7->9, 0->8, 15->3, 31->23, 23->25, 26->24, 21->23, 23->25, 32->24, 30->22, 25->23, 23->21, 21->7, 6->8, 20->6, 9->7, 11->9, 6->8, 9->7, 2->0, 0->8, 7->9, 9->1 Here are solutions to all these alternate games including another solution to start/end at 1.

Move Start/End Start/End Start/End Start/End Start/End Start/End Nbr at 0 at 1 at 3 at 4 at 8 at 9 1 2 to 0 9 to 1 5 to 3 16 to 4 0 to 8 1 to 9 2 9 to 1 7 to 9 16 to 4 1 to 9 2 to 0 16 to 4 3 0 to 2 0 to 8 1 to 9 14 to 16 5 to 3 7 to 9 4 7 to 9 2 to 0 14 to 16 3 to 15 16 to 4 0 to 8 5 10 to 8 9 to 7 3 to 15 6 to 8 3 to 5 4 to 16 6 2 to 10 6 to 8 6 to 8 9 to 7 7 to 9 11 to 9 7 11 to 9 11

 to 9 9 to 7 11 to 9 10 to 2 2 to 10 8 8 to 10 9 to 7 17 to 5 2 to 10 12 to 10 9 to 7 9 17 to 5 20 to 6 2 to 10 9 to 11 9 to 11 6 to 8 10 15 to 17 6 to 8 15 to 17 12 to 10 21 to 7 15 to 3 11 13 to 15 15 to 3 17 to 5 16 to 14 6 to 8 13 to 15 12 22 to 8 0 to 8 12 to 10 13 to 15 15 to 3 17 to 5 13 20 to 22 21 to 7 5 to 17 17 to 5 0 to 8 

22 to 8 14 23 to 21 7 to 9 20 to 6 19 to 17 20 to 6 3 to 15 15 24 to 10 16 to 4 6 to 8 22 to 8 23 to 21 15 to 17 16 5 to 17 23 to 21 27 to 15 20 to 22 24 to 10 20 to 22 17 18 to 16 24 to 10 15 to 3 24 to 10 19 to 17 23 to 21 18 26 to 24 5 to 17 0 to 8 5 to 17 26 to 24 24 to 10 19 12 to 26 18 to 16 24 to 22 26 to 24 30 to 22 5 to

 17 20 30 to 22 26 to 24 21 to 23 23 to 25 21 to 23 18 to 16 21 21 to 23 12 to 26 26 to 24 27 to 15 24 to 22 26 to 24 22 23 to 25 30 to 22 23 to 25 15 to 3 31 to 23 12 to 26 23 26 to 24 21 to 23 32 to 24 0 to 8 32 to 24 30 to 22 24 31 to 23 23 to 25 17 to 29 7 to 9 17 to 29 21 to 23 25 23 to 25 26 to 24 19 to 17 32 to 24 2

2 to 24 23 to 25 26 32 to 24 31 to 23 30 to 32 17 to 29 29 to 17 26 to 24 27 25 to 23 23 to 25 32 to 24 30 to 32 17 to 5 31 to 23 28 23 to 9 32 to 24 25 to 23 32 to 24 2 to 10 23 to 25 29 9 to 7 25 to 23 28 to 16 25 to 23 11 to 9 32 to 24 30 6 to 8 23 to 9 17 to 15 28 to 16 9 to 7 25 to 23 31 8 to 0 9 to 1 15 to 3 16 to 4 6 to 8

 23 to 9 Still another variation of the game is to leave 4 pegs in the corners of the square surrounding the center (i.e.

in positions 8, 10, 22, and 24). A sample solution is: 4->16, 7->9, 0->8, 2->0, 9->7, 6->8, 10->2, 12->10, 15->3, 0->8, 13->15, 15->3, 17->5, 2->10, 19->17, 17->15, 22->8, 3->15, 20->22, 22->8, 24->22, 26->24, 27->15, 29->27, 30->22, 15->27, 32->30, 30->22 The number of ways to solve this variation is: 4,540,128,887,763,134,208 or The following table shows all possible single peg ending holes for possible starting holes.

Initial Empty All Possible Single Hole Peg Ending Holes 0 0, 15, 18, 30 1 1, 13, 16, 19, 31 2 2, 14, 17, 32 3 3, 22, 25 4 4, 20, 23, 26 5 5, 21, 24 6 6, 9, 12, 28 7 7, 10, 29 9 6, 9, 12, 28 10 7, 10, 29 12 6, 9, 12, 28 13 1, 13, 16, 19, 31 14 2, 14, 17, 32 15 0, 15, 18, 30 16 1, 13, 16, 19, 31 17 2, 

14, 17, 32 18 0, 15, 18, 30 19 1, 13, 16, 19, 31 20 4, 20, 23, 26 21 5, 21, 24 23 4, 20, 23, 26 24 5, 21, 24 26 4, 20, 23, 26 27 8, 11, 27 28 6, 9, 12, 28 29 7, 10, 29 30 0, 15, 18, 30 31 1, 13, 16, 19, 31 32 2, 14, 17, 32 How to calculate the number of winning solutions.

A possible algorithm to count the number of solutions could be: For each of the possible first moves (There are four as noted above) For each of these four possible moves try all possible second moves. For each of these second moves, try all possible third moves. Etc. for all possible moves through 31 turns. If you wrote a computer program using this simple algorithm and had a super computer that could generate and process 1 billion of these potential games per second, it would take over 18,000 years to complete the search of the 577+ quintillion possible games. Thus, we will look for a better algorithm. There are two standard algorithms that a computer program might use to find and count solutions to the peg problem. One algorithm would be a “depth first” search while another algorithm would be a “breadth first” search. The algorithm that we gave above is a “depth first” search as it will try to search (make trial moves) as deeply as possible, and backs up only when it hits a dead end. As noted earlier, a pure “depth first” search would take many thousands of years. The depth first search can be modified to run much faster.

We note that the peg solitaire board has 33 holes. Each hole can either be empty or it can contain a peg. Thus there is a maximum of 2^33 = 8,589,934,592 combinations of holes that may or may not have a peg in them. Most of these combinations can not occur in a standard game. (If you are just solving for total game positions, a further substantial reduction in the number of combinations is realized when you adjust for rotations and reflections.) Thus we will design a computer program that keeps track of the number of solutions that are possible as it searches from each possible position. When a subsequent set of moves reaches and recognizes one of these positions, it can merely add the number of solutions found earlier without having to retrace the same search pattern. If you are solving for total possible games for the 16/16 standard game, it turns out that "only" 23,475,688 unique patterns are encountered (adjusted for the 8 rotation/reflection variations). Even if you ignore rotations/reflections, the standard game only has 187,636,299 possible board patterns.

Thus our computer program will modify the earlier "For each of ..." algorithm as follows: A stack is used to keep track of where the search is, how many solutions have been found, what trial move to try next, etc. Initially, start the brute force search using the equivalent of the nested "For each of ..." loops. Data for the current position in the search tree is kept in a stack. As each minor subtree search is completed, a record is saved in the history arrays. Each of these records contains information on how many solutions were found, plus a look-up system that can identify the particular board position and a method to rapidly find these results when needed at some future time in the search pattern. The outline for the algorithm then becomes: do { // Repeat loop until all possible combinations have been processed // There are 76 potential moves for various peg patterns. Get next trial move // Try the next one of these for the current "For each of ..." level If a potential move is found { // If one of these is a current legal move... Calculate the position's score // If applicable, try all possible rotations/reflections of the // new pattern. // Express each of these as a binary number (Use board number // pattern for bit positions).

The lowest of these is the "score" // which will be used for pattern recognition. Look this position ("score") up in the history records If found, then add the prior number of solutions to the current running total (in the stack) and continue at the top of the "do" loop Else a new pattern has been found in the search sequence. Increment the stack pointer (move down one level in the "For each of ..." sequence) } // End of "If a potential move is found" sequence. Return // to the top of the "do {" loop to continue the search. Else { // A potential move was not found. Reached the end of move // combinations. The subtree search at this level is complete. Add to history data // Create a new record for this game position in the history arrays. // If this position is seen again at some future point in the // search sequence, the relevant data is available and the // subtree sequence does not have to be searched again.

Restore the former board pattern Move up one level in the stack } // End of else not found sequence } while more stack levels // Repeat the "do" loop until the search is complete Output the number of solutions // Report the result before ending the program Execution time on modern multicore processors is typically just a few minutes. The algorithm can be modified to use any of the 33 holes as a start hole, and then try to find solutions by ending at any arbitrary finish hole. If you are just solving for total possible games, the program takes full advantage of symetry. (Rotations and reflections produce 8 variations of a unique position) A few paragraphs ago, we mentioned that the peg problem can also be solved by a “breadth first” search. In a breadth first search we start with all known positions after “N” moves, and then find all possible positions for the “N+1” move. Initially there is only one board position. (Empty space at the start position.) For the standard game, initially there is only one position (The “old” position). If we try all possible valid moves, the algorithm finds 4 possible “new” positions at the end of round 1.

These 4 new positions are then relabled as “old” positions, and all possible moves are tried using these “old” positions to see how many “new” positions can be found for the end of round 2. The computer also keeps track how how many combinations are found The processes is continued for the 31 possible rounds of play. On the 31st move the algorithm finds all possible finishing holes. If one of these is the solution location, the “remembered” number of combinations becomes the number of ways to win. Thus our computer algorithm for a breadth first becomes: Initialize the system with data for the game’s “Start position” in the “New” arrays. For 31 rounds of possible jumps Copy the board position data from the “new” arrays to the “old” arrays For each of the "old" positions (typically there are tens of millions of them) Try all possible next moves (There are 76 possible moves subject to peg pos.) Each time a move can be made, update the position data in the “new” arrays. Repeat for all old records/positions. Repeat for all 31 rounds of possible jumps Note: You are dealing with many large numbers. Watch for memory and precision problems.

The above algorithm can also be optimized. The Computer Programs Two different computer programs (using different algorithms) were used to calculate the results shown on this web page. Both programs came up with the same number of wins and same number of total possible games. (This is always comforting.) However, each program could supply a little additional information that the other couldn’t. The first program used a modified tree search (depth first) with interim results kept on a stack. The standard tree search was modified so that the program could recognize positions that had been visited before. The source code for the current version of this “depth first” search program can be seen here.

Question: How many pegs or marbles are there in a Solitaire board game?
Answer: 33